#302000

order	512
nilpotency class	3
isoclinism	isoclinic class of #301993
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1050 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,6)(4,7)(5,8)(9,16)(10,14)(11,13)(12,15)(17,18,22,25)(19,27,30,32)(20,21,26,23)(24,28,31,29);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,12)(15,16)(17,19,23,28,22,30,21,29)(18,24,26,32,25,31,20,27);(1,4)(2,7)(3,9,10,11)(5,12)(6,13,14,16)(8,15)(18,25)(19,28,30,29)(20,26)(24,27,31,32);(4,12)(7,15)(9,11)(13,16)(17,20,21,25,22,26,23,18)(19,27,29,31,30,32,28,24)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=fh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,g^a=gi,c^b=cg,f^b=fi,g^b=gh,d^c=di,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 209196);